We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in symmetric powers $C^{(e)}$ of $C$ where $3\leq e\leq g-3$. More precisely, given a pencil $g^1_d$ of degree $d$ on $C$, let $X$ be the curve parametrizing divisors of degree $e$ in divisors of $g^1_d$ (see the paper for the precise scheme-theoretical definition). Under certain genericity assumptions on the pair $(C, g^1_d)$, we prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=2e$, dim$H^0 (g^1_d) = e$ or $d=2e+1$, dim$H^0 (g^1_d) = e+1$. The analogous result in the case $e=2$ without genericity assumptions was proved earlier.
We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in the second symmetric power $C^{(2)}$ of $C$. More precisely, given a pencil $g^1_d$ of degree $d$ on $C$, let $X$ be the curve parametrizing pairs of points in divisors of $g^1_d$ (see the paper for the precise scheme-theoretical definition). We prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=4$ or $d=5$, dim$H^0 (g^1_5) = 3$ and $C$ has gen…
We prove that every polynomial map $(f,g):\mathbb{R}^2\to\mathbb{R}^2$ with nowhere vanishing Jacobian such that $\mathrm{deg}\, f\leq 5$, $\mathrm{deg}\,g \leq 6$ is injective.
In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+δ)$ where $δ$ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension $\leq 9$ in the case $\prod_{i=1}^{n}a_{ii}\neq0$.
We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and Möller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any $g \leq 4$.
In the paper, we first classify all polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(°_yu,°_yh)\leq 3$, $H(0)=0$. Then we classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,u), h(x,y))$ in the case that $JH$ is nilpotent and $(°v(x,y,0),°h)\leq 3$, $H(0)=0$. Finally, we classify polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in certain conditions.
We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree $k$ homogeneous component of its Jacobian ideal for any $d-1\leq k\leq (n+1)(d-2)$. Our results generalize the previous result on the reconstruction of a homogeneous polynomial from its Jacobian ideal.
In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the…
First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also …
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
For an algebraic variety $X$ of dimension $d$ with totally degenerate reduction over a $p$-adic field (definition recalled below) and an integer $i$ with $1\leq i\leq d$, we define a rigid analytic torus $J^i(X)$ together with an Abel-Jacobi mapping to it from the Chow group $CH^i(X)_{hom}$ of codimension $i$ algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over the complex numbers. We compare and contrast the complex and $p$-adic theories. Finally, we examine a special case of a $p…
Let $f_1,...,f_n$ be $n$ germs of holomorphic functions at the origin of $\mathbb{C}^n$ such that $f_i(0)=0$, $1\leq i\leq n$. We give a proof based on the J. Lipman's theory of residues via Hochschild Homology that the Jacobian of $f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ if and only if the dimension ot the germ of common zeos of $f_1,...,f_n$ is sttrictly positive. In fact we prove much more general results which are relatives versions of this result replacing the field $\mathbb{C}$ by convenient noetherian rings $\mathbf{A}…
Let $f=(f_1, f_2)$ be a regular sequence of affine curves in $\bC^2$. Under some reduction conditions achieved by composing with some polynomial automorphisms of $\bC^2$, we show that the intersection number of curves $(f_i)$ in $\bC^2$ equals to the coefficient of the leading term $x^{n-1}$ in $g_2$, where $n=°f_i$ $(i=1, 2)$ and $(g_1, g_2)$ is the unique solution of the equation $y{\mathcal J}(f)=g_1f_1+g_2f_2$ with $°g_i\leq n-1$. So the well-known Jacobian problem is reduced to solving the equation above. Furthermore, by using the result above, we show that the Jacobian problem can also b…
In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the dimension of the locus of hyperelliptic curves of genus $g$ with $p$-rank at most $f$. We also produce families of curves so that the $p$-torsion of the Jacobian of each fibre contains multiple copies of the group scheme $α_p$. The method is to study curves which admit an action by $(\ZZ/2)^n$ so that the quotient is a projective line. As a result, some of these fa…
We survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic) curves. For example, if $0 \leq f \leq g-1$, we prove that every component of the $p$-rank $f+1$ stratum of ${\mathcal M}_g$ contains a component of the $p$-rank $f$ stratum in its closure. We prove that the $p$-rank $f$ stratum of $\overline{\mathcal M}_g$ is connected. For all primes $p$ and all $g \geq 4$, we demonstrate the existence of a Jacobian of a smoo…
We study the $p$-rank stratification of the moduli space of Prym varieties in characteristic $p > 0$. For arbitrary primes $p$ and $\ell$ with $\ell \not = p$ and integers $g \geq 3$ and $0 \leq f \leq g$, the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified ${\mathbb Z}/\ell$-covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \geq 5$ and $\ell = 2$, the second theorem implies that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified double cover whose Prym has $p$-rank $f'$ fo…
